5-dimensional warped cosmological solutions with radius stabilization by a bulk scalar
Introduction
The Randall–Sundrum (RS) idea [1] for explaining the weak-scale hierarchy problem has garnered much attention from both the phenomenology and string-theory communities, providing a link between the two which is often absent. RS is a simple and elegant way of generating the TeV scale which characterizes the standard model from a set of fundamental scales which are of order the Planck mass (Mp). All that is needed is that the distance between a hidden and a visible sector brane be approximately b=35/Mp in a compact extra dimension, y∈[0,1]. The warping of space in this extra dimension, by a factor e−kby, translates the moderately large interbrane separation into the large hierarchy needed to explain the ratio TeV/Mp.
However the RS idea as originally proposed was incomplete due to the lack of any mechanism for stabilizing the brane separation, b. This was a modulus, corresponding to a massless particle, the radion, which would be ruled out because of its modification of gravity: the attractive force mediated by the radion would effectively increase Newton's constant at large distance scales. An attractive model for giving the radion a potential energy was proposed by Goldberger and Wise (GW) [2]; they introduced a bulk scalar field with different VEVs, v0 and v1, on the two branes. If the mass m of the scalar is small compared to the scale k which appears in the warp factor e−kby, then it is possible to obtain the desired interbrane separation. One finds the relation e−kb≅(v1/v0)4k2/m2.
An important benefit of stabilizing the radion is that cosmology is governed by the usual Friedmann equations, up to small corrections of order ρ/(TeV)4 [3]. Even with stabilization, there may be a problem with reaching a false minimum of the GW radion potential [4], but without stabilization, there is a worse problem: an unnatural tuning of the energy densities on the two branes is required for getting solutions where the extra dimension is static [5], [6], a result which can be derived using the (5,5) component of the Einstein equation Gmn=κ2Tmn. However when there is a nontrivial potential for the radius, V(b), the (5,5) equation serves only to determine the shift δb in the radius due to the expansion, and there is no longer any constraint on the matter on the branes. Although this point is now well appreciated [7], [8], [9], [10], it has not previously been explicitly demonstrated by solving the full 5-dimensional field equations using a concrete stabilization mechanism. Indeed, it has been claimed recently that such solutions are not possible with an arbitrary equation of state for the matter on the branes [12], [13], and also that the rate of expansion does not reproduce normal cosmology on the negative tension brane despite stabilization [14]. Our purpose is to present the complete solutions, to leading order in an expansion in the energy densities on the branes, thus refuting these claims.
Section snippets
Preliminaries
The action for 5-D gravity coupled to the stabilizing scalar field Φ and matter on the branes (located at y=0 and y=1, respectively) is where κ2 is related to the 5-D Planck scale M by κ2=1/(M3). The negative bulk cosmological constant needed for the RS solution is parametrized as Λ=−6k2/κ2 and the scalar field potential is that of a free field, . The brane potentials V0 and V1 can have any form that will insure
Perturbation equations
We can now write the equations for the perturbations of the metric, δA, δN, δb, and the scalar field, δΦ. The equations take a simpler form when expressed in terms of the following combinations: Further simplification comes from realizing that the perturbations will have the form, for example, , so that their time derivatives are proportional to and . Below we will confirm that , where is the Hubble parameter.
Solutions
Naively, it would appear that we have five equations for four unknown perturbations, but of course since gravity is a gauge theory, this is not the case. First, we have the relation . Furthermore, the (55) Einstein equation and the scalar equation can be shown to be equivalent, using (00), (ii) and the zeroth order relations (7): . So our system is actually underdetermined because of unfixed gauge degrees of
Stiff potential limit
The above solutions are quite general, but they are not complete because we have not yet solved for the scalar field perturbation, δΦ. This would generically be intractable, but there is a special case in which things simplify, namely, when the brane potentials Vi(Φ) become stiff. In this case, the boundary condition for the scalar fluctuation becomes δΦ=0 at either brane. There is no information about the derivative δΦ′ in this case; although δΦ→0, at the same V″(Φ)→∞ in such a way that the
Implications
Above we focused on the shift in the size of the extra dimension due to cosmological expansion, but the more experimentally relevant quantity is the shift in the lapse function, n(t,1), evaluated on the TeV brane. As emphasized in Ref. [11], the change in n(t,1) between the present and the past determines how much physical energy scales on our brane, like the weak scale, MW, have evolved. The time dependence of MW is given by MW(t)/MW(t0)=e−δN(t,1)+δN(t0,1)+δN(t,0)−δN(t0,0). In terms of the
Acknowledgements
We thank C. Csaki, M. Graesser and G. Kribs for helpful discussions. J.C. thanks Nordita for its hospitality while this work was being finished.
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